For the quantum Calogero-Moser model, we construct a set of conserved operators and another set of operators, named boost operators, from its Lax operator. We prove that each conserved operator satisfies both the Lax equation and a remarkable relation named additional relation. From these knowledge, we show that the conserved operators are involutive. Moreover, the conserved operators and the boost operators constitute the U(1)-current algebra. All the proofs are simplified a great deal due to the Lax equations and additional relations.
For any root system
Δ and an irreducible representation
R of the reflection (Weyl) group
GΔ generated by
Δ, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member
μ of
R, to be called a "site", we associate a vector space
Vμ whose element is called a "spin". Its dynamical variables are the canonical coordinates
{qj,pj} of a particle in
Rr, (
r= rank of
Δ), and spin exchange operators
{P^ρ} (
ρ∈Δ) which exchange the spins at the sites
μ and
sρ(μ). Here
sρ is the reflection generated by
ρ. For each
Δ and
R a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For
Δ=Ar and
R= vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.
Given a critical quantum spin chain with a microscopic Lie group symmetry, corresponding, e.g., to
U(1) or
SU(2) spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated with conformal invariance is augmented with a Kac-Moody algebra associated with conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low-energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allows us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.
For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a “site”, we associate a vector space Vµ whose element is called a “spin”. Its dynamical variables are the canonical coordinates {qj,pj} of a particle in R r, (r = rank of ∆), and spin exchange operators { ˆ Pρ} (ρ ∈ ∆) which exchange the spins at the sites µ and sρ(µ). Here sρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by “freezing ” the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For ∆ = Ar and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.